I think this can work and instead of that the heatpump pumps the heat into your house (when "solar is plenty") it would pump it into the storage. (I have a similar setup, but heat the water but of course this is rather limited)

unrelated: a simple technical solution to your window problem would be home assistant and a few sensors to notify you when the windows are open too long or open when too cold inside.

It’s more about predictive modeling: At what time + temperature do you close windows, given predicted cloud cover and overnight temperatures/wind?

Come to think of it, if we had a big salt slurry that transitioned at 72F in the floors, that would probably do the right thing. It’d create a step function making it hard to heat above 72, or cool below it.

I wonder how much density changes as these things transition. Would a static pool (mixed by freezing) work, or would it need a pump?